FCLoopFindTopologyMappings[{topo1, topo2, ...}] finds
mappings between topologies (written as FCTopology objects)
topo1, topo2, .... For each source topology the function
returns a list of loop momentum shifts and a GLI
replacement rule needed to map it to the given target topology. If you
need to map everything to a particular set of target topologies, you can
specify them via the PreferredTopologies option.
The output is a list of two lists, the former containing the mappings and the latter enumerating the final contributing topologies
To enable shifts in the external momenta you need to set the option
Momentum to All.
Overview, FCTopology, GLI, FCLoopFindTopologies.
Here we have a set of 5 topologies
topos1 = {
FCTopology[fctopology1, {SFAD[{{p3, 0}, {0, 1}, 1}], SFAD[{{p2, 0}, {0, 1}, 1}],
SFAD[{{p1, 0}, {0, 1}, 1}], SFAD[{{p2 + p3, 0}, {0, 1}, 1}], SFAD[{{p2 - Q, 0}, {0, 1}, 1}],
SFAD[{{p1 - Q, 0}, {0, 1}, 1}], SFAD[{{p2 + p3 - Q, 0}, {0, 1}, 1}], SFAD[{{p1 + p3 - Q, 0}, {0, 1}, 1}],
SFAD[{{p1 + p2 + p3 - Q, 0}, {0, 1}, 1}]}, {p1, p2, p3}, {Q}, {}, {}],
FCTopology[fctopology2, {SFAD[{{p3, 0}, {0, 1}, 1}],
SFAD[{{p2, 0}, {0, 1}, 1}], SFAD[{{p1, 0}, {0, 1}, 1}], SFAD[{{p2 + p3, 0}, {0, 1}, 1}],
SFAD[{{p2 - Q, 0}, {0, 1}, 1}], SFAD[{{p1 - Q, 0}, {0, 1}, 1}],
SFAD[{{p2 + p3 - Q, 0}, {0, 1}, 1}], SFAD[{{p1 + p2 - Q, 0}, {0, 1}, 1}],
SFAD[{{p1 + p2 + p3 - Q, 0}, {0, 1}, 1}]}, {p1, p2, p3}, {Q}, {}, {}],
FCTopology[fctopology3, {SFAD[{{p3, 0}, {0, 1}, 1}],
SFAD[{{p2, 0}, {0, 1}, 1}], SFAD[{{p1, 0}, {0, 1}, 1}],
SFAD[{{p2 + p3, 0}, {0, 1}, 1}], SFAD[{{p1 + p3, 0}, {0, 1}, 1}],
SFAD[{{p2 - Q, 0}, {0, 1}, 1}], SFAD[{{p2 + p3 - Q, 0}, {0, 1}, 1}],
SFAD[{{p1 + p3 - Q, 0}, {0, 1}, 1}], SFAD[{{p1 + p2 + p3 - Q, 0}, {0, 1}, 1}]},
{p1, p2, p3}, {Q}, {}, {}],
FCTopology[fctopology4, {SFAD[{{p3, 0}, {0, 1}, 1}],
SFAD[{{p2, 0}, {0, 1}, 1}], SFAD[{{p1, 0}, {0, 1}, 1}],
SFAD[{{p2 + p3, 0}, {0, 1}, 1}], SFAD[{{p1 + p3, 0}, {0, 1}, 1}],
SFAD[{{p2 - Q, 0}, {0, 1}, 1}], SFAD[{{p1 - Q, 0}, {0, 1}, 1}],
SFAD[{{p1 + p3 - Q, 0}, {0, 1}, 1}], SFAD[{{p1 + p2 + p3 - Q, 0}, {0, 1}, 1}]},
{p1, p2, p3}, {Q}, {}, {}],
FCTopology[fctopology5, {SFAD[{{p3, 0}, {0, 1}, 1}],
SFAD[{{p2, 0}, {0, 1}, 1}], SFAD[{{p1, 0}, {0, 1}, 1}],
SFAD[{{p1 + p3, 0}, {0, 1}, 1}], SFAD[{{p2 - Q, 0}, {0, 1}, 1}],
SFAD[{{p1 - Q, 0}, {0, 1}, 1}], SFAD[{{p1 + p3 - Q, 0}, {0, 1}, 1}],
SFAD[{{p1 + p2 - Q, 0}, {0, 1}, 1}], SFAD[{{p1 + p2 + p3 - Q, 0}, {0, 1}, 1}]},
{p1, p2, p3}, {Q}, {}, {}]};3 of them can be mapped to the other two
mappings1 = FCLoopFindTopologyMappings[topos1];\text{FCLoopFindTopologyMappings: }\;\text{Found }3\text{ mapping relations }
\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }2
mappings1[[1]]\left( \begin{array}{ccc} \;\text{FCTopology}\left(\text{fctopology3},\left\{\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p2}+\text{p3})^2+i \eta )},\frac{1}{((\text{p1}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p2}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right) & \{\text{p1}\to -\text{p1}-\text{p3}+Q,\text{p2}\to -\text{p2}-\text{p3}+Q\} & G^{\text{fctopology3}}(\text{n1$\_$},\text{n7$\_$},\text{n8$\_$},\text{n5$\_$},\text{n6$\_$},\text{n4$\_$},\text{n2$\_$},\text{n3$\_$},\text{n9$\_$}):\to G^{\text{fctopology1}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7},\text{n8},\text{n9}) \\ \;\text{FCTopology}\left(\text{fctopology4},\left\{\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p2}+\text{p3})^2+i \eta )},\frac{1}{((\text{p1}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right) & \{\text{p1}\to Q-\text{p2},\text{p2}\to Q-\text{p1},\text{p3}\to -\text{p3}\} & G^{\text{fctopology4}}(\text{n1$\_$},\text{n6$\_$},\text{n5$\_$},\text{n8$\_$},\text{n7$\_$},\text{n3$\_$},\text{n2$\_$},\text{n4$\_$},\text{n9$\_$}):\to G^{\text{fctopology1}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7},\text{n8},\text{n9}) \\ \;\text{FCTopology}\left(\text{fctopology5},\left\{\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p1}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right) & \{\text{p1}\to \;\text{p2},\text{p2}\to \;\text{p1}\} & G^{\text{fctopology5}}(\text{n1$\_$},\text{n3$\_$},\text{n2$\_$},\text{n4$\_$},\text{n6$\_$},\text{n5$\_$},\text{n7$\_$},\text{n8$\_$},\text{n9$\_$}):\to G^{\text{fctopology2}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7},\text{n8},\text{n9}) \\ \end{array} \right)
And these are the final topologies
mappings1[[2]]\left\{\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p2}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta )},\frac{1}{((\text{p2}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right),\text{FCTopology}\left(\text{fctopology2},\left\{\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p2}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta )},\frac{1}{((\text{p2}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right)\right\}
Here is another example
topos2 = {FCTopology[fctopology1, {SFAD[{{q2, 0}, {0, 1}, 1}],
SFAD[{{q1, 0}, {0, 1}, 1}], SFAD[{{q1 + q2, 0}, {0, 1}, 1}], SFAD[{{p + q1, 0}, {0, 1}, 1}],
SFAD[{{p - q2, 0}, {0, 1}, 1}]}, {q1, q2}, {p}, {}, {}],
FCTopology[fctopology2, {SFAD[{{q2, 0}, {0, 1}, 1}], SFAD[{{q1, 0}, {0, 1}, 1}],
SFAD[{{p + q2, 0}, {0, 1}, 1}], SFAD[{{p - q1, 0}, {0, 1}, 1}]}, {q1, q2}, {p}, {}, {}],
FCTopology[fctopology3, {SFAD[{{q2, 0}, {0, 1}, 1}], SFAD[{{q1, 0}, {0, 1}, 1}],
SFAD[{{p - q1, 0}, {0, 1}, 1}], SFAD[{{p - q1 + q2, 0}, {0, 1}, 1}]}, {q1, q2}, {p}, {}, {}]}\left\{\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((\text{q1}+\text{q2})^2+i \eta )},\frac{1}{((p+\text{q1})^2+i \eta )},\frac{1}{((p-\text{q2})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{fctopology2},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((p+\text{q2})^2+i \eta )},\frac{1}{((p-\text{q1})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{fctopology3},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((p-\text{q1})^2+i \eta )},\frac{1}{((p-\text{q1}+\text{q2})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right)\right\}
Yet this time we have some preferred set of topologies and want to match to them (if possible)
preferredTopos2 = {FCTopology[prop2L, {SFAD[{{q1, 0}, {0, 1}, 1}],
SFAD[{{q2, 0}, {0, 1}, 1}], SFAD[{{q1 - q2, 0}, {0, 1}, 1}], SFAD[{{-p + q1, 0}, {0, 1}, 1}],
SFAD[{{-p + q2, 0}, {0, 1}, 1}]}, {q1, q2}, {p}, {}, {}],
FCTopology[prop2LX1, {SFAD[{{q2, 0}, {0, 1}, 1}], SFAD[{{q1 - q2, 0}, {0, 1}, 1}],
SFAD[{{-p + q1, 0}, {0, 1}, 1}], SFAD[{{-p + q2, 0}, {0, 1}, 1}]}, {q1, q2}, {p}, {}, {}],
FCTopology[prop2LX3, {SFAD[{{q1, 0}, {0, 1}, 1}], SFAD[{{q2, 0}, {0, 1}, 1}],
SFAD[{{-p + q1, 0}, {0, 1}, 1}], SFAD[{{-p + q2, 0}, {0, 1}, 1}]}, {q1, q2}, {p}, {}, {}],
FCTopology[prop2LX15, {SFAD[{{q2, 0}, {0, 1}, 1}], SFAD[{{q1 - q2, 0}, {0, 1}, 1}],
SFAD[{{-p + q1, 0}, {0, 1}, 1}]}, {q1, q2}, {p}, {}, {}]}\left\{\text{FCTopology}\left(\text{prop2L},\left\{\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-\text{q2})^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX1},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-\text{q2})^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX3},\left\{\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX15},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-\text{q2})^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right)\right\}
mappings2 = FCLoopFindTopologyMappings[topos2, PreferredTopologies -> preferredTopos2];\text{FCLoopFindTopologyMappings: }\;\text{Found }3\text{ mapping relations }
\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }3
mappings2[[1]]\left( \begin{array}{ccc} \;\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((\text{q1}+\text{q2})^2+i \eta )},\frac{1}{((p+\text{q1})^2+i \eta )},\frac{1}{((p-\text{q2})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right) & \{\text{q1}\to -\text{q2},\text{q2}\to \;\text{q1}\} & G^{\text{fctopology1}}(\text{n1$\_$},\text{n2$\_$},\text{n3$\_$},\text{n5$\_$},\text{n4$\_$}):\to G^{\text{prop2L}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5}) \\ \;\text{FCTopology}\left(\text{fctopology2},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((p+\text{q2})^2+i \eta )},\frac{1}{((p-\text{q1})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right) & \{\text{q1}\to \;\text{q2},\text{q2}\to -\text{q1}\} & G^{\text{fctopology2}}(\text{n1$\_$},\text{n2$\_$},\text{n3$\_$},\text{n4$\_$}):\to G^{\text{prop2LX3}}(\text{n1},\text{n2},\text{n3},\text{n4}) \\ \;\text{FCTopology}\left(\text{fctopology3},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((p-\text{q1})^2+i \eta )},\frac{1}{((p-\text{q1}+\text{q2})^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right) & \{\text{q1}\to \;\text{q2},\text{q2}\to \;\text{q2}-\text{q1}\} & G^{\text{fctopology3}}(\text{n2$\_$},\text{n1$\_$},\text{n4$\_$},\text{n3$\_$}):\to G^{\text{prop2LX1}}(\text{n1},\text{n2},\text{n3},\text{n4}) \\ \end{array} \right)
And these are the final occurring topologies
mappings2[[2]]\left\{\text{FCTopology}\left(\text{prop2L},\left\{\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-\text{q2})^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX1},\left\{\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-\text{q2})^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX3},\left\{\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta )}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right)\right\}
If we need to match subtopologies into larger topologies, we first need to generate all possible subtopologies for each relevant topology.
topos3 = {
FCTopology[fctopology1, {
SFAD[{{l1 + l2 - q1, 0}, {0, 1}, 1}],
SFAD[{{l2, 0}, {SMP["m_t"]^2, 1}, 1}],
SFAD[{{l1, 0}, {SMP["m_t"]^2, 1}, 1}],
SFAD[{{l2 + q2, 0}, {SMP["m_t"]^2, 1}, 1}],
SFAD[{{l1 - q1, 0}, {SMP["m_t"]^2, 1}, 1}],
SFAD[{{l1 - q1 - q2, 0}, {SMP["m_t"]^2, 1}, 1}]}, {l1, l2}, {q1, q2}, {}, {}],
FCTopology[fctopology9, {
SFAD[{{l1 + l2 + q2, 0}, {0, 1}, 1}],
SFAD[{{l2, 0}, {SMP["m_t"]^2, 1}, 1}],
SFAD[{{l1, 0}, {SMP["m_t"]^2, 1}, 1}],
SFAD[{{l1 + q2, 0}, {SMP["m_t"]^2, 1}, 1}],
SFAD[{{l1 - q1, 0}, {SMP["m_t"]^2, 1}, 1}]}, {l1, l2}, {q1, q2}, {}, {}]
}\left\{\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{((\text{l1}+\text{l2}-\text{q1})^2+i \eta )},\frac{1}{(\text{l2}^2-m_t^2+i \eta )},\frac{1}{(\text{l1}^2-m_t^2+i \eta )},\frac{1}{((\text{l2}+\text{q2})^2-m_t^2+i \eta )},\frac{1}{((\text{l1}-\text{q1})^2-m_t^2+i \eta )},\frac{1}{((\text{l1}-\text{q1}-\text{q2})^2-m_t^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\{\},\{\}\right),\text{FCTopology}\left(\text{fctopology9},\left\{\frac{1}{((\text{l1}+\text{l2}+\text{q2})^2+i \eta )},\frac{1}{(\text{l2}^2-m_t^2+i \eta )},\frac{1}{(\text{l1}^2-m_t^2+i \eta )},\frac{1}{((\text{l1}+\text{q2})^2-m_t^2+i \eta )},\frac{1}{((\text{l1}-\text{q1})^2-m_t^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\{\},\{\}\right)\right\}
subTopos3 = Flatten[FCLoopFindSubtopologies[topos3]];subTopos3 // Length37
Now we can match a smaller topology into a larger topology
mappings3 = FCLoopFindTopologyMappings[topos3, PreferredTopologies -> subTopos3];\text{FCLoopFindTopologyMappings: }\;\text{Found }1\text{ mapping relations }
\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }1
mappings3[[1]]\left( \begin{array}{ccc} \;\text{FCTopology}\left(\text{fctopology9},\left\{\frac{1}{((\text{l1}+\text{l2}+\text{q2})^2+i \eta )},\frac{1}{(\text{l2}^2-m_t^2+i \eta )},\frac{1}{(\text{l1}^2-m_t^2+i \eta )},\frac{1}{((\text{l1}+\text{q2})^2-m_t^2+i \eta )},\frac{1}{((\text{l1}-\text{q1})^2-m_t^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\{\},\{\}\right) & \{\text{l1}\to \;\text{q1}-\text{l1},\text{l2}\to -\text{l2}-\text{q2}\} & G^{\text{fctopology9}}(\text{n1$\_$},\text{n3$\_$},\text{n4$\_$},\text{n5$\_$},\text{n2$\_$}):\to G^{\text{fctopology1}}(\text{n1},0,\text{n2},\text{n3},\text{n4},\text{n5}) \\ \end{array} \right)
mappings3[[2]]\left\{\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{((\text{l1}+\text{l2}-\text{q1})^2+i \eta )},\frac{1}{(\text{l2}^2-m_t^2+i \eta )},\frac{1}{(\text{l1}^2-m_t^2+i \eta )},\frac{1}{((\text{l2}+\text{q2})^2-m_t^2+i \eta )},\frac{1}{((\text{l1}-\text{q1})^2-m_t^2+i \eta )},\frac{1}{((\text{l1}-\text{q1}-\text{q2})^2-m_t^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\{\},\{\}\right)\right\}
Mapping the following two topologies onto each other requires shifts in the external momenta due to the chosen kinematic constraints.
topos4 = {
FCTopology[topo1, {
SFAD[{{l1 + q1, 0}, {m^2, 1}, 1}],
SFAD[{{l1 - l2, 0}, {0, 1}, 1}],
SFAD[{{l2 + q1, 0}, {m^2, 1}, 1}],
SFAD[{{l2 - q2, 0}, {m^2, 1}, 1}],
SFAD[{{l2, 0}, {0, 1}, 1}]}, {l1, l2}, {q1, q2}, {SPD[q1, q1] -> 0, SPD[q2, q2] -> 0, SPD[q1, q2] -> s/2}, {}],
FCTopology[topo2, {
SFAD[{{l1 - l2, 0}, {m^2, 1}, 1}],
SFAD[{{l1 - q2, 0}, {0, 1}, 1}],
SFAD[{{l2 - q2, 0}, {m^2, 1}, 1}],
SFAD[{{l2 + q1, 0}, {m^2, 1}, 1}],
SFAD[{{l2, 0}, {0, 1}, 1}]}, {l1, l2}, {q1, q2}, {SPD[q1, q1] -> 0, SPD[q2, q2] -> 0, SPD[q1, q2] -> s/2}, {}]}\left\{\text{FCTopology}\left(\text{topo1},\left\{\frac{1}{((\text{l1}+\text{q1})^2-m^2+i \eta )},\frac{1}{((\text{l1}-\text{l2})^2+i \eta )},\frac{1}{((\text{l2}+\text{q1})^2-m^2+i \eta )},\frac{1}{((\text{l2}-\text{q2})^2-m^2+i \eta )},\frac{1}{(\text{l2}^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\left\{\text{q1}^2\to 0,\text{q2}^2\to 0,\text{q1}\cdot \;\text{q2}\to \frac{s}{2}\right\},\{\}\right),\text{FCTopology}\left(\text{topo2},\left\{\frac{1}{((\text{l1}-\text{l2})^2-m^2+i \eta )},\frac{1}{((\text{l1}-\text{q2})^2+i \eta )},\frac{1}{((\text{l2}-\text{q2})^2-m^2+i \eta )},\frac{1}{((\text{l2}+\text{q1})^2-m^2+i \eta )},\frac{1}{(\text{l2}^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\left\{\text{q1}^2\to 0,\text{q2}^2\to 0,\text{q1}\cdot \;\text{q2}\to \frac{s}{2}\right\},\{\}\right)\right\}
mappings4 = FCLoopFindTopologyMappings[topos4, Momentum -> All];\text{FCLoopFindTopologyMappings: }\;\text{Found }1\text{ mapping relations }
\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }1
mappings4[[1]]\left( \begin{array}{ccc} \;\text{FCTopology}\left(\text{topo2},\left\{\frac{1}{((\text{l1}-\text{l2})^2-m^2+i \eta )},\frac{1}{((\text{l1}-\text{q2})^2+i \eta )},\frac{1}{((\text{l2}-\text{q2})^2-m^2+i \eta )},\frac{1}{((\text{l2}+\text{q1})^2-m^2+i \eta )},\frac{1}{(\text{l2}^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\left\{\text{q1}^2\to 0,\text{q2}^2\to 0,\text{q1}\cdot \;\text{q2}\to \frac{s}{2}\right\},\{\}\right) & \{\text{l1}\to -\text{l1}+\text{l2}-\text{q1},\text{q1}\to -\text{q2},\text{q2}\to -\text{q1}\} & G^{\text{topo2}}(\text{n1$\_$},\text{n2$\_$},\text{n3$\_$},\text{n4$\_$},\text{n5$\_$}):\to G^{\text{topo1}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5}) \\ \end{array} \right)
Otherwise no mappings exist
FCLoopFindTopologyMappings[topos4][[1]]\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies topo2 and topo1. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}
\text{FCLoopFindTopologyMappings: }\;\text{Found }0\text{ mapping relations }
\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }2
\{\}
Topologies containing eikonal or other nonstandard propagators may introduce additional challenges. Even though two such topologies can be recognized to be identical, the code still would not be able to work out the correct momentum shifts without some additional input.
topoEik1 = FCTopology[mytopo67, {SFAD[{{k2, 0}, {0, 1}, 1}], SFAD[{{k1, 0}, {0, 1}, 1}],
SFAD[{{k1 + k2, 0}, {0, 1}, 1}], SFAD[{{0, -k1 . nb}, {0, 1}, 1}],
SFAD[{{k2, -meta u0b k2 . nb}, {0, 1}, 1}], SFAD[{{k1 + k2, -2 gkin meta u0b (k1 + k2) . n},
{0, 1}, 1}], SFAD[{{k1, -2 gkin meta k1 . n + meta u0b k1 . nb}, {2 gkin meta^2 u0b, 1}, 1}]},
{k1, k2}, {n, nb}, {Hold[SPD][n] -> 0, Hold[SPD][nb] -> 0, Hold[SPD][n, nb] -> 2}, {}];topoEik2 = FCTopology[mytopo79, {SFAD[{{k2, 0}, {0, 1}, 1}], SFAD[{{k1, 0}, {0, 1}, 1}],
SFAD[{{0, k1 . nb}, {0, 1}, 1}], SFAD[{{k2, -meta u0b k2 . nb}, {0, 1}, 1}],
SFAD[{{k1 + k2, -meta u0b (k1 + k2) . nb}, {0, 1}, 1}], SFAD[{{k1,
2 gkin meta k1 . n - meta u0b k1 . nb}, {2 gkin meta^2 u0b, 1}, 1}],
SFAD[{{k1 + k2, 2 gkin meta u0b (k1 + k2) . n - meta u0b (k1 + k2) . nb},
{2 gkin meta^2 u0b^2, 1}, 1}]}, {k1, k2}, {n, nb}, {Hold[SPD][n] -> 0,
Hold[SPD][nb] -> 0, Hold[SPD][n, nb] -> 2}, {}];DataType[gkin, FCVariable] = True;
DataType[meta, FCVariable] = True;
DataType[u0b, FCVariable] = True;At first sight these two topologies are independent from each other
FCLoopFindTopologyMappings[{topoEik1, topoEik2}];\text{FCLoopFindMomentumShifts: }\;\text{The topologies contain following mixed quadratic-eikonal propagators that complicate the determination of the shifts: }\left\{\frac{1}{(\text{k1}^2+\text{k1}\cdot (2 \;\text{gkin} \;\text{meta} n-\text{meta} \;\text{u0b} \;\text{nb})-2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}+i \eta )},\frac{1}{(\text{k1}^2+\text{k1}\cdot (\text{meta} \;\text{u0b} \;\text{nb}-2 \;\text{gkin} \;\text{meta} n)-2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}+i \eta )},\frac{1}{(\text{k2}^2-\text{meta} \;\text{u0b} (\text{k2}\cdot \;\text{nb})+i \eta )},\frac{1}{((\text{k1}+\text{k2})^2-2 \;\text{gkin} \;\text{meta} \;\text{u0b} ((\text{k1}+\text{k2})\cdot n)+i \eta )},\frac{1}{((\text{k1}+\text{k2})^2-\text{meta} \;\text{u0b} ((\text{k1}+\text{k2})\cdot \;\text{nb})+i \eta )},\frac{1}{((\text{k1}+\text{k2})^2+(\text{k1}+\text{k2})\cdot (2 \;\text{gkin} \;\text{meta} \;\text{u0b} n-\text{meta} \;\text{u0b} \;\text{nb})-2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}^2+i \eta )}\right\}
\text{FCLoopFindMomentumShifts: }\;\text{You can try to trade them for purely quadratic propagators using FCLoopReplaceQuadraticEikonalPropagators.}
\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies mytopo79 and mytopo67. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}
\text{FCLoopFindTopologyMappings: }\;\text{Found }0\text{ mapping relations }
\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }2
However, if we tell the code how some eikonal propagators can be brought into a quadratic form, then an explicit mapping can be found
toposNew = FCLoopReplaceQuadraticEikonalPropagators[{topoEik1, topoEik2},
LoopMomenta -> {k1, k2},
InitialSubstitutions -> {
ExpandScalarProduct[SPD[k1 - k2]] -> SPD[k1 - k2],
ExpandScalarProduct[SPD[k1 + k2]] -> SPD[k1 + k2]},
IntermediateSubstitutions -> {SPD[n] -> 0, SPD[nb] -> 0, SPD[n, nb] -> 2}];```mathematica eikMappings = FCLoopFindTopologyMappings[toposNew];
```mathematica
\text{FCLoopFindTopologyMappings: }\;\text{Found }1\text{ mapping relations }
\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }1